They'd probably need to be persecuted far more harshly. While they do have a martyring complex which would drive faith, and they did genuinely have moments they were persecuted, this persecution was sporadic and not as hard as it could have been.
A decent example of a way to effectively have them flee east would be to have them get persecuted as badly as Manichaeism was when the Christians gained supremacy. After all, for a time the Manichees were rivals in influence to the Christians until they were legitimized. IE you'd need to enact an empire wide pogrom that sets out to kill priests and evict people similar to that and make being a christian illegal.
And this attitude would definitely make the Sassanids more inclined to support the christians; they supported the Lahkmids for being members of the Church of the East precisely to use as a counterpoint to Rome.